How to use the measurement uncertainty budget calculator
- List each source of uncertainty with a name.
- Choose its type: Normal (an expanded uncertainty with its coverage factor, as on a calibration certificate), Rectangular or Triangular (the half width of the limits), or Standard (a standard uncertainty already).
- Enter the value and, where it applies, the coverage factor k and the sensitivity coefficient c that scales the source into the result.
- Set the coverage factor for the final result, and read the combined and expanded uncertainty and the share of each source.
Formula
Normal source: u = value / k. Rectangular source: u = value / sqrt(3). Triangular source: u = value / sqrt(6)
Contribution of each source = c x u
Combined standard uncertainty u_c = square root of the sum of the squared contributions
Expanded uncertainty U = coverage factor x u_c
This is the root sum of squares method of the Guide to the Expression of Uncertainty in Measurement for uncorrelated sources. The share column is each squared contribution divided by the total.
A calibration and a resolution limit
A calibration certificate gives a standard uncertainty of 0.10, and the instrument resolution is a rectangular limit of 0.20. The resolution contributes 0.20 / sqrt(3) = 0.11547. The combined standard uncertainty is sqrt(0.10^2 + 0.11547^2) = sqrt(0.023333) = 0.152753.
With a coverage factor of 2 the expanded uncertainty is 0.305505. The calibration accounts for 42.86 percent of the variance and the resolution for 57.14 percent, so resolution is the largest contributor.
Type A and type B
A type A source is evaluated from repeated readings, and its standard uncertainty is the standard error of the mean. A type B source is evaluated from other information, such as a calibration certificate, a specification limit or a resolution. Both end up as a standard uncertainty, which is why they can be combined.
For the standard error of repeated readings, use the standard error calculator. To carry uncertainties through a formula with several measurements, use the error propagation calculator.
Reporting the result
State the result with its expanded uncertainty, the coverage factor and the level it represents, and round the uncertainty to one or two significant figures. The significant figures calculator helps with the rounding. A coverage factor of 2 corresponds to about 95 percent for a normal distribution with many degrees of freedom.
Assumptions and limits
- Sources are independent. Correlated sources need covariance terms that this page does not include.
- The coverage factor applies to a roughly normal result. With few effective degrees of freedom a larger factor may be needed.
- Rectangular and triangular limits are the half widths of the range.
- The budget is only as good as the list of sources. Missing a major source understates the result.
Frequently asked questions
What is the difference between standard and expanded uncertainty?
Standard uncertainty is one standard deviation. Expanded uncertainty multiplies it by a coverage factor to give an interval with a stated level of confidence.
Why divide by the square root of 3?
A rectangular distribution with a half width of a has a standard deviation of a divided by the square root of 3. It is used when all you know is the limits.
What is a sensitivity coefficient?
It is how much the result changes when a source changes by one unit. It is 1 when the source adds directly to the result.
Does this replace an accredited uncertainty evaluation?
No. It supports the arithmetic of a budget. The choice of sources and distributions is your judgment, and a formal evaluation needs documented evidence.